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Chapter 1 Chapter 1 Functions and Limits Functions and Limits § 1 Functions § Functions I . Definition 2. Set ( 集合) Def: A collection of numbers which have the same property. natural number rational number (positive, negative) integer 自然数(正、负)整数有理数 1. The Four Arithmetic Operations (四则运算) add sum subtract difference multiply divide product quotient (加减乘除) (和差积商) § Functions Some expressions: negative positive a times b a over b numerator denominator a plus b a minus b even number odd number § Functions I . Definition 3. Intervals (区间) Open Interval }{bxax??),(ba?oxab }{bxax??],[ba?oxab }{bxax??],(ba?oxab Closed Interval ),(}{bbxx ????oxb ),[}{ ????aaxxoxa ),( ?????Rx Infinite Interval § Functions I . Definition 4. Neighborhood of point a (点 a的邻域)x a??a??a }0{),(:??????axxaU Def o Geometric Interpretation: The set of point x which distance from point a is less than .? a is the center of this neighborhood, and is the radius. ?}|{}{),(:????????????axaxaxxaU Def§ Functions II . Functions 1. Definition A function is a rule of correspondence that associate with each object x in one set and a single value y from a second set. )(xf y RyDx f???????? Domain Range independent variable dependent variable f§ Functions II . Functions Q: how to find the domain of a function? x xy 1 arctan 3 1???5 7 12 os sin 2 3 2????xx x- yxy Q: how to discriminate two functions are the same one? xxgxxf lg2)( lg)( 2??,)()(xfytfu??, 1. Definition § Functions II . Functions 2. Special Functions absolute value function ????????0, 0xx x x, xyx o y greatest integer function ][x = the greatest integer less than or equal to o y sign function ???????????0,1 0,0 0,1 sgn x x xxyxxx?? sgnx o y § Functions III . Properties of Functions 1. Boundedness W D xf????)( Def: ,)(,0,,MxfMXxDX?????? that such if Then we say f (x ) is bounded on X. ),)(()(,0,MxfM

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  • 时间2016-04-20